Mental Health America was a faltering education and advocacy nonprofit in 2014 when its new president, Paul Gionfriddo, launched a quick project: the creation of an online screening tool.

The tool would help users evaluate whether they were showing symptoms of various mental-health conditions and direct them to resources for help. Mental Health America had long believed that it should help older people who had lived with mental illness for decades. But staff were surprised to find that most people who did the screening were women under 35.

That experience carries a lesson for all organizations that seek to modernize their work, Mr. Gionfriddo says: "Make sure you really know your constituents — not who you think they are."

Such new thinking is just one of the many hallmarks of sweeping change at Mental Health America. Once slow-moving and bureaucratic, the group now embraces experimentation, social media, and novel hiring approaches. "We’re 100-something years old," Mr. Gionfriddo says. "But we’re functioning like a start-up."